The deterministic version of the Old-Keynesian model The estimations begin with the most elementary sentiment dynamics (SD–1) without

4.2. The deterministic version of the Old-Keynesian model The estimations begin with the most elementary sentiment dynamics (SD–1) without

any random forces. An obvious steady state position is given by y = 0, π = π ⋆ and i = i . For the continuous-time model with the simplified Taylor rule (µ i = 0), in Franke (2011a)

a broad range of parameters with a sufficiently high herding coefficient φ b (in the present notation) was shown to exist that ensure uniqueness of this equilibrium, render it locally repelling, and give rise to a unique and globally attractive limit cycle. This feature carries over to the present formulation in discrete time and the slightly more general Taylor rule

(9). Such a limit cycle can be viewed as the model’s representative business cycle. 28 Our first MSM estimation then searches for a numerical parameter combination such that the

78 autocovariances from the list in (4) that it induces are as close as possible to their empirical counterparts. The solution of the corresponding minimization problem (1) with J=J (78) is reported as Scenario SD–1a in Table 1.

Before trying to assess whether a minimal loss of J (78) = 71.45 is more indicative of

a good or a bad match, we should have a look at the cycles thus generated. The time series of the output gap y t and the inflation rate π t are shown in the upper two panels in Figure 1. With a bit of more than five years, they may perhaps exhibit acceptable amplitudes and an acceptable period, but the pattern of the cyclical motions is clearly unsatisfactory. Output in reality simply does not crawl along a ceiling for roughly two years, then suddenly drops down on a floor and proceeds creeping there for another two years. It is thus also superfluous to comment on the tent-shape pattern of the inflation rate.

Responsible for this behaviour is the fact that the constraints in the transition prob- abilities, prob − +

, prob +− t ≤ 1 in (12), become binding over these stages. If we look at the specification of the feedback index f t in eq. (13) then, owing to the high values of the (stabilizing) coefficient φ i on the real rate of interest and the (destabilizing) herding

coefficient φ b ,f t actually becomes so large in modulus that it takes quite a while for it to return to more moderate values; and if it eventually does, it only takes two or three quarters until, with signs reversed, f t soars to similarly high levels again. Apart from the unrealistic time series pattern, the resulting extreme probabilities are not very convincing, either.

These observations suggest introducing an additional moment m 79 into the objective function for the estimations. It considers the, as we may call them, “excess transition probabilities” α b exp(±f t ) −1 from (12) and penalizes the occurrence of positive values

28 To be more precise, there is a unique one-dimensional manifold P in the three-dimensional space towards which all (non-degenerate) trajectories converge in the sense that they move on P

in the limit, although the limit motion itself may not be strictly periodic but only quasi-periodic.

Scenario Model SD

Model NK 1a 1b 2a 2b 3a 3b a b

Table 1: Estimations of models SD and NK (Great Inflation). Note : Bold face figures indicate the type of loss function for which the scenario is optimal. High

values of J (79) are truncated at 999. Underlying NK–a, NK–b and the four stochastic scenarios of SD is the same random seed ¯ c: among 1000 estimations with different random seeds, this ¯ c yields the median loss J (82) for Scenario 3b.

so heavily that the loss minimization procedure better seeks to avoid them, even at the cost of a worse match of the other 78 moments. Formally we proceed in three steps.

First, m 79 is specified as the average excess transition probability over the simulation horizon S (with respect to a given time path of the feedback index f t , which in turn, of course, depends on the parameters θ and possibly a random seed c); second, its empirical counterpart is set equal to zero, to conform to the notation of the loss function (1); and third, we incorporate the penalty in the new diagonal element of the now (79×79)

Figure 1: Time series y t ,π t resulting from Scenario SD–1a and SD–1b. weighting matrix W , for which a value of 1000 turns out to be perfectly suitable. Thus,

m 79 = max{ 0, α b exp(f t ) − 1 } + max{ 0, α b exp(−f t )−1} S t =1

=0 (15) W 79,79 = 1000 The correspondingly augmented loss function is designated J (79) ,

m emp

T ;79,79

J (79) : loss function constituted by the 78 autocovariances and weights (16)

from (2), (4), plus moment m 79 with weight W 79,79 from (15) Applying the new function J (79) to Scenario SD–1a quantifies its deficient time series

features; the value we compute is so unacceptably high that in Table 1 we arbitrarily truncated it at 999.

The re-estimation of the deterministic model with J (79) , which forms our Scenario 1b, confirms that the model can do better. As the table shows, adding the new criterion somewhat deteriorates the original matching, i.e. the loss J (78) from the autocovariances increases from 71.45 to 75.15. However, this seems a relatively low price for the total success concerning the excess transition probabilities, which have practically vanished (the 79th component of the loss is practically zero). Comparing the parameters in Sce- narios 1a and 1b it is seen that the general improvement is essentially brought about

by considerably lower values of the two sentiment parameters φ b and φ i . They neverthe- less balance in a certain way, such that the (repelling) instability of the steady state is maintained. The smooth oscillations of Scenario 1b that we obtain are documented in by considerably lower values of the two sentiment parameters φ b and φ i . They neverthe- less balance in a certain way, such that the (repelling) instability of the steady state is maintained. The smooth oscillations of Scenario 1b that we obtain are documented in

On this sound basis we can now ask what is behind the pure number of the minimized loss J (79) = 75.15. Figure 2 presents the profiles of the nine auto- and cross-covariances of the three single variables i t ,y t ,π t over a lag horizon of 20 quarters, which—it may

be taken into account—is longer than the eight quarters underlying the estimations themselves. The empirical covariances are given by the dotted lines with the shaded area of a 95% confidence band around them. 29 The profiles generated by Scenario 1b are plotted as the thin (blue) lines. At a glance and taking the confidence bands as

a guideline, the matching can already be reckoned quite satisfactory. While there is a certain moderate tendency to leave the confidence band at the higher lags, within the first eight lags we observe only three cases with stronger deviations in this respect, all of which at a zero lag. That is, these are the simulated variances of the interest rate in the upper-left panel, of the output gap in the central panel, and of the inflation rate in the lower-right panel, where all of these moments are too low.

Table 2 reports selected t-statistics of our different estimations. With respect to Sce- nario 1b it shows that, in terms of this criterion, the first two “violations” are not very serious. The most critical point of the deterministic model is rather its inability to match the variance of the inflation rate or, more precisely, to trace out the sudden drop of Cov(π t ,π t−h ) from h = 0 to h = 1; since afterwards the changes in the autocovariances remain relatively limited, the estimation decides to “sacrifice” the matching of the zero lag. It will have to be seen if the stochastic versions of the model can fare better in this respect.

As a secondary aspect we note in Table 2 that there is essentially one moment with which Scenario 1b pays for the more appropriate time series pattern vis-` a-vis Scenario 1a. This is the variance of the output gap with its deterioration to t i = −2.30, which the estimation of Scenario 1a had managed to keep inside the confidence interval. Regarding the autocovariances of the inflation rate, Scenario 1a and 1b share the same problem.